Linear lower bounds for δc(p) for a class of 2D self‐destructive percolation models

The self‐destructive percolation model is defined as follows: Consider percolation with parameter p > p_c. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance δ to become occupied. Let δ_c(p) be the minimal value of δ, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced by van den Berg and Brouwer. They showed, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) that δ_c(p) ≥ . In particular, δ_c(p) is at least linear in p ? p_c. Although the arguments used by van den Berg and Brouwer look very lattice‐specific, we show that they can be suitably modified to obtain similar linear lower bounds for δ_c(p) (with p near p_c) for a much larger class of 2D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n,p) for np = 1 + ε, where ε = ε(n) = o(1). We show that if ε³n →—∞, then a.a.s. D(n,p) consists of components which are either isolated vertices or directed cycles, each of size O_p(|ε|^?1). On the other hand, if ε³n →∞, then a.a.s. the structure of D(n,p) is dominated by the unique complex component of size (4 + o(1))ε²n, whereas all other components are of size O_p(ε^?1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

一类2n+1次多项式微分系统的局部极限环分支

研究了一类2n 1次多项式微分系统在原点的局部极限环分支问题,通过计算与理论推导得出了该系统原点的奇点量表达式,确定了系统原点的中心条件以及最高阶细焦点的条件,并在此基础上构造出系统在原点分支出4个极限环的实例.     

Global existence of solutions for 2‐D semilinear wave equations with dissipation localized near infinity in an exterior domain

We shall derive some global existence results to semilinear wave equations with a damping coefficient localized near infinity for very special initial data in H×L². This problem is dealt with in the two‐dimensional exterior domain with a star‐shaped complement. In our result, a power p on the non‐linear term |u|^p is strictly larger than the two‐dimensional Fujita‐exponent. Copyright © 2005 John Wiley & Sons, Ltd.     

Signed and sign-changing solutions for a Kirchhoff-type problem involving the fractional p-Laplacian with critical Hardy nonlinearity

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets.     

Bifurcation of limit cycles at a nilpotent critical point in a septic Lyapunov system

In this paper, we characterize local behavior of an isolated nilpotent critical point for a class of septic polynomial differential systems, including center conditions and bifurcation of limit cycles. With the help of computer algebra system-MATHEMATICA 12.0, the first 15 quasi-Lyapunov constants are deduced. As a result, necessary and sufficient conditions of such system having a center are obtained. We prove that there exist 16 small amplitude limit cycles created from the third-order nilpotent critical point. And then we give a lower bound of cyclicity of third-order nilpotent critical point for septic Lyapunov systems.     

Existence and multiplicity of weak solutions for a class of fractional Sturm-Liouville boundary value problems with impulsive conditions

In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional differential equations with non-homogeneous Sturm-Liouville conditions and impulsive conditions by using the critical point theory. In addition, at the end of this paper, we also give the existence results of infinite weak solutions of fractional differential equations under homogeneous Sturm-Liouville boundary value conditions. Finally, several examples are given to illustrate our main results.     

Dynamics of a viral infectiology under treatment

This paper deals with a nonlinear model of the viral dynamics which describes the interactions between the human immune system and the virus. The novelty of this work is the introduction of combined treatments in the dynamics to modify the model. We investigate the qualitative behavior of the model and find a threshold parameter that guarantees the asymptotic stability of the equilibrium points, this parameter is known as the basic reproduction number. We estimated the parameters of the model by least-squares minimization between the numerical solution of the system and clinical data of cell cultures. It is also demonstrated that critical drug efficacy in terms of the model parameter is greatly useful to curtail the spreading of the disease.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.